Most of the existing feasibility results on Byzantine Agreement (BA) are of an all-or-nothing fashion: in Broadcast they address the question whether or not there exists a protocol which allows any player to broadcast his input. Similarly, in Consensus the question is whether or not consensus can be reached which respects pre-agreement on the inputs of all correct players. In this work, we introduce the natural notion of player-centric BA which is a class of BA primitives, denoted as , parametrized by subsets of the player set. For each primitive the validity is defined on the input(s) of the players in . Broadcast (with sender p) and Consensus are special (extreme) cases of primitives for and , respectively. We study feasibility of in the presence of a general (aka non-threshold) mixed (active/passive) adversary, and give a complete characterization for perfect, statistical, and computational security. Our results expose an asymmetry of Broadcast which has, so far, been neglected in the literature: there exist non-trivial adversaries which can be tolerated for Broadcast with sender some but not for some other being the sender. Finally, we extend the definition of by adding fail corruption to the adversary's capabilities, and give exact feasibility bounds for computationally secure (aka Consensus) in this setting. This answers an open problem from ASIACRYPT 2008 concerning feasibility of computationally secure multi-party computation in this model.